Question

Прошу вас , помогите !!!!!!!!! Дам 50 баллов

Answer 1

Ответ:

1.

$\int\limits^{ \frac{}{5} }_{0} \cos {}^{2} (5x)dx = \int\limits^{ \frac{\pi}{5} }_{0} \frac{1 + \cos(10x) }{2}dx = \\ = \frac{1}{2} ( \int\limits^{ \frac{\pi}{5} }_{0} dx + \frac{1}{10} \int\limits^{ \frac{\pi}{5} }_{0} \cos(10x)d(10x)) = \\ = \frac{1}{2} (x + \frac{1}{10} \sin(10x)) \ | ^{ \frac{\pi}{5} }_{0} = \\ = \frac{1}{2} ( \frac{\pi}{5} + \frac{1}{10} \sin(2\pi) - 0 - 0) = \frac{\pi}{10}$

2.

$\int\limits^{9}_{4} \frac{x + 1}{ \sqrt{x} + 2 } dx \\ \\ \\ \sqrt{x} = t \\ x = t {}^{2} \\ dx = 2tdt \\ t1 = \sqrt{9} = 3\\ t2 = \sqrt{4} = 2 \\ \\ 2\int\limits^{3}_{2} \frac{ {t}^{2} + 1 }{t + 2} tdt = 2\int\limits^{3}_{2}( {t}^{2} - 2t + 5 - \frac{10}{t + 2} )dx = \\ =2 ( \frac{ {t}^{3} }{3} - {t}^{2} + 5t - 10 ln( |t + 2| ) ) | ^{3}_{2} = \\ = 2(9 - 9 + 45 - 10 ln(5) - \frac{8}{3} + 4 - 10 + 10 ln(4) ) = \\ = \frac{38}{3} - 20 ln( \frac{5}{4} )$